Secure function evaluation for a covert client and a semi-honest server using string selection oblivious transfer

ABSTRACT

Methods and apparatus are provided for secure function evaluation for a covert client and a semi-honest server using string selection oblivious transfer. An information-theoretic version of a garbled circuit C is sliced into a sequence of shallow circuits C 1 , . . . C n , that are evaluated. Consider any wire w j  of C that is an output wire of C i , and is an input wire of C i+1 . When a slice C i  is evaluated, C i &#39;s 1-bit wire key for w j  is computed by the evaluator, and then used, via string selection oblivious transfer (SOT), to obtain the wire key for the corresponding input wire of C i+1 . This process repeats until C&#39;s output wire keys are computed by the evaluator. The 1-bit wire keys of the output wires of the slice are randomly assigned to wire values.

STATEMENT OF GOVERNMENT RIGHTS

The present invention was made in connection with work performed under the Intelligence Advanced Research Projects Activity (IARPA) via Department of the Interior (DOI) Contract No. D11PC20194.

FIELD OF THE INVENTION

The present invention relates generally to techniques for securing electronic transactions and, more particularly, to secure function evaluation (SFE) techniques that provide privacy to the parties of such electronic transactions.

BACKGROUND OF THE INVENTION

Two-party general secure function evaluation (SFE) allows two parties to evaluate any function on their respective inputs x and y, while maintaining the privacy of both x and y. Efficient SFE algorithms enable a variety of electronic transactions, previously impossible due to mutual mistrust of participants. For example, SFE algorithms have been employed in auctions, contract signing and distributed database milling applications. As computation and communication resources have increased, SEE has become truly practical for common use. A malicious SFE model provides a guarantee of complete privacy of the players inputs, even when a dishonest player follows an arbitrary cheating strategy.

Existing generic two-party SFE algorithms typically employ Garbled Circuits (GCs). For a detailed discussion of GCs, see, for example, Y. Lindell and B. Pinkas, “A Proof of Yao's Protocol for Secure Two-Party Computation,” Journal of Cryptology, 22(2):161-188 (2009). For reasonably complex functions, however, the data transfer required for SFE is prohibitive. In fact, the communication complexity of GC-based SFE protocols is dominated by the size of the GC, which can reach Megabytes or Gigabytes even for relatively small and simple functions (e.g., the GC for a single secure evaluation of the block cipher AES has size 0.5 Megabytes).

While transmission of large amounts of data is often possible, existing networks will not scale should SFE be widely deployed. This is particularly true for wireless networks, or for larger scale deployment, of secure computations, e.g., by banks or service providers, with a large number of customers. Additional obstacles include energy consumption required to transmit/receive the data, and the resulting reduced battery life in mobile clients, such as smartphones. Computational load on the server is also a significant problem. Moreover, security against more powerful malicious adversaries requires the use of the standard cut-and-choose technique, which in turn requires transfer of multiple GCs.

A need remains for improved techniques for secure function evaluation where both parties are assumed to be semi-honest.

SUMMARY OF THE INVENTION

Generally, methods and apparatus are provided for secure function evaluation for a covert client and a semi-honest server using string selection oblivious transfer. According one aspect of the invention, a server performs secure function evaluation with a client by obtaining a circuit, C, representing a function, f; preparing slices of the circuit, C, into a sequence of sub-circuits C₁, . . . C_(l), wherein each of the sub-circuits C₁, . . . C_(l) comprises a fan-out-one circuit; executing an oblivious transfer of keys for the sub-circuits C₁, . . . C_(l), to the client for evaluation, wherein input keys of one or more of the sub-circuits C₁, . . . C_(l), are based on output keys of a prior one of the sub-circuits C₁, . . . C_(l), wherein a secret associated with the oblivious transfer is only sent to the client if one of a plurality of allowed selection strings is provided by the client; and sending one or more output translation tables to the client.

According another aspect of the invention, a client performs secure function evaluation with a server by executing an oblivious transfer of keys for a plurality of sub-circuits C₁, . . . C_(l), from the server for evaluation, wherein the sub-circuits C₁, . . . C_(l) are a plurality of information-theoretic garblings of slices of a circuit, C, wherein each of the sub-circuits C₁, . . . C_(l) comprises a fan-out-one circuit, and wherein input keys of one or more of the sub-circuits C₁, . . . C_(l), are based on output keys of a prior one of the sub-circuits C₁, . . . C_(l), wherein a secret associated with the oblivious transfer is only received by the client if one of a plurality of allowed selection strings is sent by the client to the server; evaluating the sub-circuits C₁, . . . C_(l); using garbled input values to obtain garbled output values; receiving one or more output translation tables from the server; and generating a bit for each output wire of the circuit, C, corresponding to a wire secret obtained in evaluation of the sub-circuit C_(l).

According yet another aspect of the invention, a server performs a string-selection oblivious transfer of one or more secrets with a client, wherein the server holds two pairs of secrets (x₀, r₀), (x₁, r₁), wherein x₀ and x₁ comprise k-bit strings and wherein r₀ and r₁ comprise k′-bit strings, a wherein r₀≠r₁, by encrypting the secret x₀ with a key s₀, and the secret x₁ with a key s₁ to provide two encryptions e₀ and e₁, wherein the keys s₀ and s₁ are chosen randomly; participating in oblivious transfer with the client by executing k′ standard oblivious transfers, two oblivious transfer input secrets of the server being the two corresponding i-th secret-shares of keys s₀ and s₁, wherein the secret sharing of s₀ and s₁ takes into account two server input values r₀ and r₁; and sending the two encryptions e₀ and e₁ to the client if the client successfully reconstructed one of the two keys s₀ and s₁.

According a further aspect of the invention, a client performs a string-selection oblivious transfer of one or more secrets with a server, wherein the client holds a k-bit selection string r, by participating in oblivious transfer with the server by executing k′ standard oblivious transfers, wherein an i-th oblivious transfer is executed with a client selection bit corresponding to an i-th bit of an input selection string r; reconstructing an encryption key corresponding to the selection string r; providing to the server an indication of a successful reconstruction of one of two keys s₀ and s₁; receiving two encryptions e₀ and e₁; and decrypting one of the two encryptions e₀ and e₁ to obtain one of (x₀, x₁), according to the selection string r.

A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an improved system for generic two-party SFE that can implement the processes of the present invention;

FIG. 2 is a block diagram of Garbled Circuit-based SFE;

FIG. 3 is a flow chart describing an exemplary implementation of an SFE process that incorporates aspects of the present invention; and

FIG. 4 is a flow chart describing an exemplary implementation of a string-selection oblivious transfer (SOT) function incorporating aspects of the present invention.

DETAILED DESCRIPTION

Embodiments of the present invention improve upon existing generic two-party SFE algorithms using an information-theoretic version of garbled circuits (GC). According to one aspect of the invention, an information-theoretic version of a garbled circuit C is sliced into a sequence of shallow circuits C₁, . . . C_(n), that are evaluated.

In a semi-honest model, secure piece-wise secure circuit evaluation can be achieved as follows. Consider any slicing of C, where some wire w_(j) of C is an output wire of C_(i), and is an input wire of C_(i+1). Now, when a slice C_(i) is evaluated, C_(i)'s 1-bit wire key for w_(j) is computed by the evaluator, and then used, via oblivious transfer (OT), to obtain the wire key for the corresponding input wire of C_(i+1). This process repeats until C's output wire keys are computed by the evaluator. In order to prevent the evaluator from learning the intermediate wire values of C, the 1-bit wire keys of slices' output wires are randomly assigned to wire values.

Further aspects of the present invention recognize that while the above semi-honest embodiment is secure against passive (semi-honest) adversaries, it can be compromised by an active evaluator, also referred to as a covert client. Indeed, the covert client can influence the output of the computation by flipping the 1-bit key of w_(j) before using it in oblivious transfer, which will result in flipping the underlying bit on that wire. To efficiently resolve this problem and achieve covert security against the evaluator, a variant of the 1-out-of-2 oblivious transfer is introduced, referred to as String-selection OT (SOT). Under String-selection OT, the receiver submits a selection string (rather than just one bit), where the OT secret held by the server is sent only if one of the two allowed strings is presented by the client. The use of the selection string prevents a cheating client from flipping the wire value, since the client now would not know which string corresponds to the opposite wire value.

FIG. 1 is a block diagram of an improved generic two-party SFE system. 100 that cam implement the features of the present invention. As shown in FIG. 1, memory 130 configures the processor 120 to implement the generic two-party SFE methods, steps, and functions disclosed herein (collectively, shown as 150 in FIG. 1). The generic two-party SFE methods 150 further comprise a GC generator 300, discussed further below in conjunction with FIG. 3. The memory 130 could be distributed or local and the processor 120 could be distributed or singular. The memory 130 could be implemented as an electrical, magnetic or optical memory, or any combination of these or other types of storage devices. It should be noted that each distributed processor that makes up processor 120 generally contains its own addressable memory space. It should also be noted that some or all of computer system 100 can be incorporated into a personal computer, laptop computer, handheld computing device, application-specific circuit or general-use integrated circuit.

Generic Two-Party SFE Algorithms

As previously indicated, existing generic two-party SFE algorithms typically employ Garbled Circuits (GCs). Y. Lindell and B. Pinkas, “A Proof of Yao's Protocol for Secure Two-Party Computation,” Journal of Cryptology, 22(2):161-188 (2009). For a detailed discussion of exemplary existing generic two-party SFE algorithms, see, for example, Payman Mohassel and Matthew Franklin, “Efficiency Tradeoffs for Malicious Two-Party Computation,” Moti Yung et al., editors, PKC 2006: 9th International Conference on Theory and Practice of Public Key Cryptography, Vol. 3958 of Lecture Notes in Computer Science, 458-473 (New York, N.Y.; Apr. 24-26, 2006); Yehuda Lindell and Benny Pinkas, “An Efficient Protocol for Secure Two-Party Computation in the Presence of Malicious Adversaries,” Moni Naor, editor, Advances in Cryptology—EUROCRYPT 2007, Vol. 4515 of Lecture Notes in Computer Science, 52-78 (Barcelona, Spain, May 20-24, 2007); David P. Woodruff, “Revisiting the Efficiency of Malicious Two-Party Computation,” In Moni Naor, editor, Advances in Cryptology—EUROCRYPT 2007, Vol. 4515 of Lecture Notes in Computer Science, 79-96 (Barcelona, Spain, May 20-24, 2007); Stanislaw Jarecki and Vitaly Shmatikov, “Efficient Two-Party Secure Computation on Committed Inputs,” In Moni Naor, editor, Advances in Cryptology—EUROCRYPT 2007, Vol. 4515 of Lecture Notes in Computer Science, 97-114 (Barcelona, Spain, May 20-24, 2007); and/or Vipul Goyal et al., “Efficient Two Party and Multiparty Computation Against Covert Adversaries.” In Nigel P. Smart, editor, Advances in Cryptology—EUROCRYPT 2008, Vol. 4965 of Lecture Notes in Computer Science, 289-306 (Istanbul, Turkey, Apr. 13-17, 2008), each incorporated by reference in its entirety.

Secure Two-Party Computation

Garbled Circuits (GC)

FIG. 2 is a block diagram of a Garbled Circuit-based SFE 200. Generally, as shown in FIG. 2, Yao's Garbled. Circuit (GC) approach allows two parties, sender S with private input y and receiver R (also referred to as client C) with private input x, to securely evaluate a boolean circuit C on (x, y) without revealing any information other than the result z=C(x, y) of the evaluation. In particular, no intermediate values are revealed.

The circuit constructor S creates a garbled circuit {tilde over (C)} from the circuit C: for each wire w_(i) of C, two garblings {tilde over (w)}_(i) ⁰, {tilde over (w)}_(i) ¹ are randomly chosen, where {tilde over (w)}_(i) ^(j) is the garbled value of W_(i)'s value j. (Note: {tilde over (w)}_(i) ^(j) does not reveal j). Further, for each gate G_(i), S creates a garbled table {tilde over (T)}_(i) with the following property: given a set of garbled values of G_(i)'s inputs, {tilde over (T)}_(i) allows to recover the garbled value of the corresponding G_(i)'s output. S sends these garbled tables, together called garbled circuit {tilde over (C)}, to evaluator (receiver R).

Additionally, R (obliviously) obtains the garbled inputs {tilde over (w)}_(i) corresponding to the inputs of both parties: the garbled inputs {tilde over (y)} corresponding to the inputs y of S are sent directly: {tilde over (y)}_(i)={tilde over (y)}_(i) ^(y) ^(i) . For each of R's inputs x_(i), both parties run a 1-out-of-2 Oblivious Transfer (OT) protocol (e.g., Y. Ishai et al., “Extending Oblivious Transfers Efficiently,” Advances in Cryptology—CRYPTO 2003 (August 2003), D. Boneh, Ed., Lecture Notes in Computer Science, Vol. 2729, Springer, 145-161 (2003); or Jesper Buus Nielsen, “Extending Oblivious Transfers Efficiently—How to get Robustness Almost for Free,” Cryptology ePrint Archive, Report 2007/215 (2007), where S inputs {tilde over (x)}_(i) ⁰, {tilde over (x)}_(i) ¹ and R inputs x_(i). The OT protocol ensures that R receives only the garbled value corresponding to his or her input hit, i.e., {tilde over (x)}_(i)={tilde over (x)}_(i) ^(x) ^(i) , while S learns nothing about x_(i).

Now, R evaluates the garbled circuit {tilde over (C)} (which comprises the garbled tables {tilde over (T)}_(i)) on the garbled inputs to obtain the garbled output {tilde over (z)} by evaluating {tilde over (C)} gate by gate, using the garbled tables {tilde over (T)}_(i). Finally, R determines the plain value z corresponding to the obtained garbled output value using an output translation table sent by S.

Embodiments of the invention provide a protocol for securely computing a function ƒ, represented by a circuit C, where semi-honest S has input y and a covert client C has input x.

For a more detailed discussion of Yao's Garbled Circuit approach, see, for example, A. C.-C. Yao, “How to Generate and Exchange Secrets (Extended Abstract),” FOCS, 162-67, (1986), incorporated by reference herein.

Efficient Information-Theoretic GC for Shallow Circuits (GESS)

V. Kolesnikov, “Gate Evaluation Secret Sharing and Secure One-Round Two-Party Computation,” Advances in Cryptology—ASIACRYPT 2005 (December 2005), B. K. Roy, Ed., Lecture Notes in Computer Science, Vol. 3788, 136-55 (2005), incorporated by reference herein, describes a Gate Evaluation Secret Sharing (GESS) scheme, which is an efficient information-theoretic analog of GC. The discloses techniques perform encryption with bitwise XOR functions and bit shufflings, rather than with standard primitives such as ABS. Thus, GESS is considered more efficient than standard GC, both in computation and communication, for shallow circuits.

Generally, GESS is a secret sharing scheme, designed to match with the gate function g, as follows. The output wire keys are the secrets, from which the constructor produces four secret shares, one for each of the wire keys of the two input wires. GESS guarantees that a combination of shares, corresponding to any of the four possible gate inputs, reconstructs the corresponding key of the output wire. This secret sharing can be applied recursively, enabling reduction of SFE to oblivious transfer (to transfer the wire secrets on the input wires).

V. Kolesnikov, “Gate Evaluation Secret Sharing and Secure One-Round Two-Party Computation,” provides a SFE protocol for a boolean formula F of communication complexity≈Σd_(i) ², where d_(i) is the depth of the i-th leaf of F. This improvement provides better performance than standard GC by evaluating thin slices of the circuit. Other Information-Theoretic GC variants can also be used, with corresponding performance disadvantage.

Covert Client

Consider a semi-honest Server, and a Client-adversary that may deviate from the protocol specification in an attempt to cheat. While cheating attempts may be successful, the covert model guarantees that any attempted cheating is caught with a certain minimal probability c. See, e.g., Y. Aumann and Y. Lindell, “Security Against Coven Adversaries: Efficient Protocols for Realistic Adversaries,” TCC 2007: 4th Theory of Cryptography Conference (February 2007), S. P. Vadhan, Ed., Lecture Notes in Computer Science, Vol. 4392, 137-56.

Aumann and Lindell give three formalizations of this notion. Consider the strong explicit-cheat formulation. Informally, this variant of the covert-security definition guarantees that, if caught, the adversary does not learn the honest player's input. If not caught, the adversary may succeed either in learning the honest player's input or influencing the computation (or both). The definition is given in the standard ideal-real paradigm. Intuitively, the difference with the malicious model is that the covert ideal world allows the cheat request: if successful (based on the coin flip by a trusted party), the adversary is allowed to win, if not, honest players are informed of the cheat attempt.

Notation

Let κ be the computational security parameter. The disclosed SFE protocol is given a circuit C which represents a function ƒ that a server S (with input y) and a client C (with input x) wish to compute. Let d denote the depth of C. The disclosed SFE protocol proceeds by dividing the circuit C into horizontal slices. Let l denote the number of such slices, and let C₁, . . . , C_(l) denote these slices of C. d′ denotes the depth of each slice C_(i).

In circuit slice C_(i), let u_(i,j) (resp. v_(i,j)) denote the jth input (resp. output) wire. For a wire u_(i,j) (resp. v_(i,j)), the garbled values corresponding to 0 and 1 are referred to by ũ_(i,j) ⁰, ũ_(i,j) ¹ (resp. {tilde over (v)}_(i,j) ⁰, {tilde over (v)}_(i,j) ¹) respectively. In the disclosed protocol, let k denote the length of input wire garblings. While evaluating the garbled circuit, C will possess only one of two garbled values for each wire in the circuit. Let ũ_(i,j)′ (resp. {tilde over (v)}_(i,j)′) denote the garbled value on wire u_(i,j) (resp. v_(i,j)) that is possessed by C.

A primitive SOT_(k,k′) discussed below in conjunction with FIGS. 3 and 4 that requires a receiver (client) C, with a k′-bit selection string as input, to select one of two k-bit strings held by sender S. One embodiment of a protocol for SOT_(k,k′) uses calls to the standard 1-out-of-2 OT primitive (where receiver C, with a selection bit as input, selects one of two k-bit: strings held by sender S).

Protocol

FIG. 3 is a flow chart describing an exemplary implementation of a secure function evaluator 300 incorporating aspects of the present invention. One embodiment of the exemplary garbled circuit generator 300 of FIG. 3 uses oblivious transfer (OT), the standard 1-out-of-2 OT protocol and String-selection OT, SOT_(k,k′), a SOT protocol, as defined below. The exemplay protocol 300 is described for secure two-party computation against a semi-honest server and a covert receiver in a hybrid model with ideal access to String-selection OT (SOT), defined below:

The exemplary String-selection OT, SOT_(k,k′), has the following functionality:

Inputs: S holds two pairs (x₀, r₀),(x₁, r₁), where each x₀, x₁ are k-bit strings, and r₀, r₁ are k′-bit strings (with r₀≠r₁). Client C holds k′-bit selection string r.

Outputs: If r=r_(i) for some iε{0,1}, then client C outputs x_(i), and S outputs empty string λ. Otherwise, client C and S both output error symbol ⊥.

Circuit C is evaluated slice-by-slice. Further, each slice is viewed as a fan-out-1 circuit, as needed for the GESS scheme discussed above.

1. Circuit Preparation:

(a) Slicing. Given d,d′, server S divides circuit C of depth d into horizontal sub-circuit layers, or slices, C₁, . . . , C_(i) of depth d′.

(b) Preparing each slice. In this step, S randomly generates the output secrets of the slice, and, applying the GESS sharing scheme, obtains corresponding input secrets, as follows. Denote by u_(i,j) input wires and by v_(i,j) the output wires of the slice C_(i). For each wire v_(i,j), S picks two random garblings {tilde over (v)}_(i,j) ⁰, {tilde over (v)}_(i,j) ¹ of length k′>1 (conditioned on {tilde over (v)}_(i,j) ⁰≠{tilde over (v)}_(i,j) ¹). S then (information-theoretically) computes the GESS garblings for each input wire in the sub-circuit, as described above. Let k be the maximal length of the garblings ũ_(i,j) ⁰, ũ_(i,j) ¹ of the input wires u_(i,j). Recall, GESS does not use garbled tables.

2. Circuit Evaluation:

For 1≦i≦l, in round i do:

(a) Oblivious transfer of keys.

i. For the top slice (the sub-circuit C₁), do the standard SFE garblings transfer:

For each client input wire u_(i,j) (representing the bits of X's input x), S and client C execute a 1-out-of-2 OT protocol, where S plays the role of a sender, with inputs ũ_(i,j) ⁰, ũ_(i,j) ¹, and client C plays the role of the receiver, directly using his inputs for OT.

For each server input wire u_(i,j) (representing the bits of Σ's input y), S sends to client C one of ũ_(i,j) ⁰, ũ_(i,j) ¹, corresponding to its input bits.

ii. For slices C_(i), i≠1, transfer next slice's input keys based on the current output keys:

For each input wire u_(i,j) of the slice C_(i), S uniformly at random chooses a string r_(i,j) of length k (this will mask the transferred secrets). S then acts as sender in SOT_(k,k′) with input ((ũ_(i,j) ⁰⊕r_(i,j), {tilde over (v)}_(i−1,j) ⁰), (ũ_(i,j) ¹⊕r_(i,j), {tilde over (v)}_(i−1,j) ¹)), and client C acts as receiver with input {tilde over (v)}_(i−1,j)′ (this is the output wire secret of the one-above slice C_(i−1), which client C obtains by executing Step 2 b described next). Let client C obtain ũ_(i,j)′⊕r_(i,j) as the output from SOT_(k,k′). Once all SOT_(k,k′) had completed, S sends all r_(i,j) to client C, who then computes all ũ_(i,j)′.

(b) Evaluating the slice. C evaluates the GESS sub-circuit C_(i), using garbled input values ũ_(i,j)′ to obtain the output values {tilde over (v)}_(i,j)′.

3. Output of the Computation. Recall, without loss of generality, only C receives the output. S now sends the output translation tables to C. For each output wire of C, C outputs the bit corresponding to the wire secret obtained in evaluation of the last slice C_(l).

It can be shown that the above Protocol embodiment is secure against a semi-honest server S and a covert client C.

In the present setting with a covert client (k′=2, deterrent ⅓), C is divided into l slices, for example, each of depth d′=4, κ≈4800 the key length on input wires of each slice.

Another aspect of the invetion provides an efficient string-selection OT (SOT) protocol, secure against a semi-honest sender and a malicious receiver, which works for selection strings of arbitrary length k′. The exemplary SOT protocol is built from k′ standard 1-out-of-2 OTs. Generally, each of the two secrets x₀, x₁ is encrypted with corresponding randomly selected keys s₀, s₁. The keys s₀, s₁ are then split into k′ random shares s_(i) ^(j) (i.e. 2k′ total shares), with the restriction that the sets of s_(i) ^(j), indexed by each of the two selection strings, reconstructs the corresponding secret. The two secrets x₀, x₁ are then sent to the receiver (client) each encrypted with the corresponding key s₀, s₁. Then, in the semi-honest model, performing k′ standard OTs allows the receiver (client) to reconstruct one of the two encryption keys, corresponding to his or her selection string. This, in turn allows the receiver (client) to reconstruct the corresponding SOT secret.

To protect against a malicious or covert receiver, assume that underlying OTs are secure against such receiver. Further, allow the sender to confirm that the receiver indeed received one of the two secrets, as follows. Denote by h₀ (resp. h₁) the hash of the vector of secret shares corresponding to the secret key s₀ (resp. s₁). To confirm that receiver obtained shares to reconstruct at least one of s₀, s₁ (and hence one of x₀, x₁), the sender will send h₀⊕h₁ to the client C, and expect to receive both h₀ and h₁ back (actually, it is sufficient to receive just one of h₀, h₁ selected by client C at random).

FIG. 4 is a flow chart describing an exemplary implementation of a string-selection oblivious transfer (SOT) function 400 incorporating aspects of the present invention. As shown in FIG. 4, the exemplary SOT function 400 (SOT_(k,k′)).

The sender S has input (x₀, r₀), (x₁, r₁) with |x₀|=|x₁|=k, and |r₀|=|r₁|=k′. Let receiver (client) C have input rε{r₀, r₁}. Let κ be a security parameter, OT be a standard 1-out-of-2 OT, and H: {0,1}′→{0,1}^(κ) be a random oracle.

1. Let r₀=r₀₁, r₀₂, . . . , r_(0k′), and r₁=r₁₁, r₁₂, . . . , r_(1k′), where r_(ij) are bits. S chooses s_(i) ^(j)ε_(R) {0,1}^(κ), for iε{0,1}, jε{1, . . . k′}. Client C sets keys s₀=⊕_(i)s_(j) ^(r) ^(0,j) , and s_(j)=⊕_(j)s_(j) ^(r) ^(ij) .

2. Client C and S participate in k′ OTs as follows. For j=1 to k′:

-   -   (a) S with input (s₁ ⁰, s_(j) ¹), and client. C with input r_(j)         send their inputs to OT. Client C receives s_(j) ^(r) ^(j) .

3. S computes hashes of shares corresponding to the two secrets/selection strings: h₀=H(s₁ ^(r) ⁰¹ , s₂ ^(r) ⁰² , . . . , s_(k) ^(r) ^(0k′) ), and h₁=H(s₁ ^(r) ¹¹ , s₂ ^(r) ¹² , . . . , s_(k′) ^(r) ^(1k′) ). S then sends h=h₀⊕h₁ to client C.

4. Client C computes h_(c)=H(s_(r) ₁ , . . . , s_(r) _(k′) ) and sends to S either h_(c) or h⊕h_(c), equiprobably.

5. S checks that the hash received from client C is equal to either h₀ or h₁. If so, it sends, in random order, H(s₀)⊕x₀, H(s₁)⊕x₁ and terminates with output λ; if not, S sends ⊥ to client C and outputs failure symbol ⊥.

6. Client C computes key s_(r)=⊕_(j)s_(r) _(j) and recovers the secret by canceling out H (s_(r)).

In one exemplary implementation, redundancy can be added to x_(i) which allows client C to identify the recovered secret. The output length of H is sufficiently stretched when used to mask secrets x_(i).

It can be shown that SOT is secure in the presence of a semi-honest sender and a malicious receiver (e.g., a covert client).

System and Article of Manufacture Details

While FIG. 3 shows an exemplary sequence of steps, it is also an embodiment of the present invention that the sequence may be varied. Various permutations of the algorithm are contemplated as alternate embodiments of the invention.

While exemplary embodiments of the present invention have been described with respect to processing steps in a software program, as would be apparent to one skilled in the art, various functions may be implemented in the digital domain as processing steps in a software program, in hardware by circuit elements or state machines, or in combination of both software and hardware. Such software may be employed in, for example, a digital signal processor, application specific integrated circuit, micro-controller, or general-purpose computer. Such hardware and software may be embodied within circuits implemented within an integrated circuit.

Thus, the functions of the present invention can be embodied in the form of methods and apparatuses for practicing those methods. One or more aspects of the present invention can be embodied in the form of program code, for example, whether stored in a storage medium, loaded into and/or executed by a machine, or transmitted over some transmission medium, wherein, when the program code is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the invention. When implemented on a general-purpose processor, the program code segments combine with the processor to provide a device that operates analogously to specific logic circuits. The invention can also be implemented in one or more of an integrated circuit, a digital signal processor, a microprocessor, and a micro-controller.

As is known in the art, the methods and apparatus discussed herein may be distributed as an article of manufacture that itself comprises a computer readable medium having computer readable code means embodied thereon. The computer readable program code means is operable, in conjunction with a computer system, to carry out all or some of the steps to perform the methods or create the apparatuses discussed herein. The computer readable medium may be a tangible recordable medium (e.g., floppy disks, hard drives, compact disks, memory cards, semiconductor devices, chips, application specific integrated circuits (ASICs)) or may be a transmission medium (e.g., a network comprising fiber-optics, the world-wide web, cables, or a wireless channel using time-division multiple access, code-division multiple access, or other radio-frequency channel). Any medium known or developed that can store information suitable for use with a computer system may be used. The computer-readable code means is any mechanism for allowing a computer to read instructions and data, such as magnetic variations on a magnetic media or height variations on the surface of a compact disk.

The computer systems and servers described herein each contain a memory that will configure associated processors to implement the methods, steps, and functions disclosed herein. The memories could be distributed or local and the processors could be distributed or singular. The memories could be implemented as an electrical, magnetic or optical memory, or any combination of these or other types of storage devices. Moreover, the term “memory” should be construed broadly enough to encompass any information able to be read from or written to an address in the addressable space accessed by an associated processor. With this definition, information on a network is still within a memory because the associated processor can retrieve the information from the network.

It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention. 

We claim:
 1. A method for secure function evaluation between a client and a server, comprising: obtaining a circuit, C, representing a function, ƒ; preparing slices of said circuit, C, into a sequence of sub-circuits C₁, . . . C_(l), wherein each of said sub-circuits C₁, . . . C_(l) comprises a fan-out-one circuit; executing an oblivious transfer of keys for said sub-circuits C₁, . . . C_(l), to said client for evaluation, wherein input keys of one or more of said sub-circuits C₁, . . . C_(l), are based on output keys of a prior one of said sub-circuits C₁, . . . C_(l), wherein a secret associated with said oblivious transfer is only sent to said client if one of a plurality of allowed selection strings is provided by said client to said server, wherein said allowed selection strings comprise a plurality of bits; and sending one or more output translation tables to said client.
 2. The method of claim 1, wherein said step of preparing slices further comprises the steps of generating output secrets of a given sub-circuit and obtaining corresponding input secrets.
 3. The method of claim 2, wherein said step of obtaining corresponding input secrets applies a Gate Evaluation Secret Sharing (GESS) sharing scheme.
 4. The method of claim 2, wherein said given sub-circuit C_(i), has input wires u_(i,j) and output wires v_(i,j), and wherein for each wire v_(i,j), said server S selects two random output garblings {tilde over (v)}_(i,j) ⁰, {tilde over (v)}_(i,j) ¹ of length k′>1 (conditioned on {tilde over (v)}_(i,j) ⁰≠{tilde over (v)}_(i,j) ¹) and computes garblings for each input wire in said given sub-circuit C_(i).
 5. The method of claim 1, wherein said step of executing said oblivious transfer of keys comprises a transfer of garblings for secure function evaluation for said sub-circuit C₁.
 6. The method of claim 5, wherein said transfer of garblings for secure function evaluation for said sub-circuit C₁ comprises said server S executing for each client input wire u_(i,j) representing bits of input x of said client C, a 1-out-of-2 oblivious transfer protocol, where S has input values ũ_(i,j) ⁰, ũ_(i,j) ¹, and said client C uses input x for said oblivious transfer.
 7. The method of claim 5, wherein said secure function evaluation garblings transfer for said sub-circuit C₁ comprises said server S sending to said client C for each server input wire u_(1,j) representing bits of input y of said server S, one of input values ũ_(i,j) ⁰, ũ_(i,j) ¹, corresponding to input bits of said server S.
 8. The method of claim 1, wherein said step of executing said oblivious transfer of keys for said sub-circuits C₂, . . . C_(l), comprises performing said oblivious transfer with server input values ((ũ_(i,j) ⁰⊕r_(i,j), {tilde over (v)}_(i−1,j) ⁰), (ũ_(i,j) ¹⊕r_(i,j), {tilde over (v)}_(i−1,j) ¹)), where r_(i,j) are randomly chosen bitmasks.
 9. The method of claim 8, further comprising the step of sending said r_(i,j) values to said client C.
 10. A method for secure function evaluation between a client and a server, comprising: executing an oblivious transfer of keys for a plurality of sub-circuits C₁, . . . C_(l), from said server for evaluation, wherein said sub-circuits C₁, . . . C_(l) are a plurality of information-theoretic garblings of slices of a circuit, C, wherein each of said sub-circuits C₁, . . . C_(l) comprises a fan-out-one circuit, and wherein input keys of one or more of said sub-circuits C₁, . . . C_(l), are based on output keys of a prior one of said sub-circuits C₁, . . . C_(l), wherein a secret associated with said oblivious transfer is only received by said client if one of a plurality of allowed selection strings is sent by said client to said server, wherein said allowed selection strings comprise a plurality of bits; evaluating said sub-circuits C₁, . . . C_(l) using garbled input values to obtain garbled output values; receiving one or more output translation tables from said server; and generating a bit for each output wire of said circuit, C, corresponding to a wire secret obtained in evaluation of said sub-circuit C_(l).
 11. The method of claim 10, wherein said step of executing said oblivious transfer of keys for said plurality of sub-circuits C₁, . . . C_(l), comprises a transfer of garblings for secure function evaluation for a sub-circuit C₁.
 12. The method of claim 10, wherein said step of executing said oblivious transfer of keys for said sub-circuits C₂, . . . C_(l), comprises performing said oblivious transfer with client input values {tilde over (v)}_(i−1,j)′, where {tilde over (v)}_(i−1,j)′ comprises an output wire secret of a prior sub-circuit C_(i−1).
 13. The method of claim 12, further comprising the step of obtaining ũ_(i,j) ¹⊕r_(i,j) as an output from a string oblivious transfer routine, SOT_(k,k′).
 14. The method of claim 13, further comprising the steps of receiving said r_(i,j) values from said server and computing said ũ_(i,j)′ values.
 15. A server system for secure function evaluation with a client, comprising: a memory; and at least one hardware device, coupled to the memory, operative to: obtain a circuit, C, representing a function, ƒ; prepare slices of said circuit, C, into a sequence of sub-circuits C₁, . . . C_(l), wherein each of said sub-circuits C₁, . . . C_(l) comprises a fan-out-one circuit; execute an oblivious transfer of keys for said sub-circuits C₁, . . . C_(l), to said client for evaluation, wherein input keys of one or more of said sub-circuits C₁, . . . C_(l), are based on output keys of a prior one of said sub-circuits C₁, . . . C_(l), wherein a secret associated with said oblivious transfer is only sent to said client if one of a plurality of allowed selection strings is provided by said client to said server, wherein said allowed selection strings comprise a plurality of bits; and send one or more output translation tables to said client.
 16. A client system for secure function evaluation with a server, comprising: a memory; and at least one hardware device, coupled to the memory, operative to: execute an oblivious transfer of keys for a plurality of sub-circuits C₁, . . . C_(l), from said server for evaluation, wherein said sub-circuits C₁, . . . C_(l) are a plurality of information-theoretic garblings of slices of a circuit, C, wherein each of said sub-circuits C₁, . . . C_(l) comprises a fan-out-one circuit, and wherein input keys of one or more of said sub-circuits C₁, . . . C_(l), are based on output keys of a prior one of said sub-circuits C₁, . . . C_(l), wherein a secret associated with said oblivious transfer is only received by said client if one of a plurality of allowed selection strings is sent by said client to said server, wherein said allowed selection strings comprise a plurality of bits; evaluate said sub-circuits C₁, . . . C_(l) using garbled input values to obtain garbled output values; receive one or more output translation tables from said server; and generate a bit for each output wire of said circuit, C, corresponding to a wire secret obtained in evaluation of said sub-circuit C_(l).
 17. A method for executing string-selection oblivious transfer of one or more secrets between a server and a client, wherein said server holds two pairs of secrets (x₀, r₀), (x₁, r₁), wherein x₀ and x₁ comprise k-bit strings and wherein r₀ and r₁ comprise k′-bit strings, and wherein r₀≠r₁, said method comprising: encrypting said secret x₀ with a key s₀, and said secret x₁ with a key s₁ to provide two encryptions e₀ and e₁, wherein said keys s₀ and s₁ are chosen randomly; participating in oblivious transfer with said client by executing k′ standard oblivious transfers, two oblivious transfer input secrets of said server being the two corresponding i-th secret-shares of keys s₀ and s₁, wherein the secret sharing of s₀ and s₁ takes into account two server input values r₀ and r₁; and sending said two encryptions e₀ and e₁ to said client if said client successfully reconstructed one of said two keys s₀ and s₁.
 18. The method of claim 17, further comprising the step of S outputing an empty string λ if r=r_(i) and said client did not deviate from a protocol.
 19. The method of claim 17, further comprising the step of outputing an error symbol ⊥ if r≠r_(i) or if said client deviated from a protocol.
 20. A method for performing oblivious transfer of one or more secrets between a server and a client, wherein said client holds a k-bit selection string r, said method comprising: participating in said oblivious transfer with said server by executing k′ standard oblivious transfers, wherein an i-th oblivious transfer is executed with a client selection bit corresponding to an i-th bit of an input selection string r; reconstructing an encryption key corresponding to said selection string r; providing to said server an indication of a successful reconstruction of one of two keys s₀ and s₁; receiving two encryptions e₀ and e₁; and decrypting one of said two encryptions e₀ and e₁ to obtain one of (x₀, x₁), according to said selection string r.
 21. The method of claim 20, further comprising the step of outputing an error symbol ⊥ if r≠r_(i) or if said client deviated from a protocol. 